%0 Conference Proceedings
%T Sparse Nerves in Practice
%+ Department of Mathematics [Bergen] (UiB)
%A Blaser, Nello
%A Brun, Morten
%< avec comité de lecture
%( Lecture Notes in Computer Science
%B 3rd International Cross-Domain Conference for Machine Learning and Knowledge Extraction (CD-MAKE)
%C Canterbury, United Kingdom
%Y Andreas Holzinger
%Y Peter Kieseberg
%Y A Min Tjoa
%Y Edgar Weippl
%I Springer International Publishing
%3 Machine Learning and Knowledge Extraction
%V LNCS-11713
%P 272-284
%8 2019-08-26
%D 2019
%R 10.1007/978-3-030-29726-8_17
%K Sparse nerve
%K Persistent homology
%K Čech complex
%K Rips complex
%Z Computer Science [cs]Conference papers
%X Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to the wide-spread use of persistent homology is its computational complexity. In order to be able to calculate persistent homology of large datasets, a number of approximations can be applied in order to reduce its complexity. We propose algorithms for calculation of approximate sparse nerves for classes of Dowker dissimilarities including all finite Dowker dissimilarities and Dowker dissimilarities whose homology is Čech persistent homology.All other sparsification methods and software packages that we are aware of calculate persistent homology with either an additive or a multiplicative interleaving. In dowker_homology, we allow for any non-decreasing interleaving function $$\alpha $$.We analyze the computational complexity of the algorithms and present some benchmarks. For Euclidean data in dimensions larger than three, the sizes of simplicial complexes we create are in general smaller than the ones created by SimBa. Especially when calculating persistent homology in higher homology dimensions, the differences can become substantial.
%G English
%Z TC 5
%Z TC 12
%Z WG 8.4
%Z WG 8.9
%Z WG 12.9
%2 https://inria.hal.science/hal-02520067/document
%2 https://inria.hal.science/hal-02520067/file/485369_1_En_17_Chapter.pdf
%L hal-02520067
%U https://inria.hal.science/hal-02520067
%~ IFIP-LNCS
%~ IFIP
%~ IFIP-TC
%~ IFIP-TC5
%~ IFIP-WG
%~ IFIP-TC12
%~ IFIP-WG8-4
%~ IFIP-WG8-9
%~ IFIP-CD-MAKE
%~ IFIP-WG12-9
%~ IFIP-LNCS-11713